The influence of source size and sampling volume on the concentration pdf of conservative tracers released in heterogeneous formations

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Advances in Water Resources 31 (2008) 339–354

Effects of pore-scale dispersion, degree of heterogeneity, samplingsize, and source volume on the concentration moments of

conservative solutes in heterogeneous formations

Daniele Tonina a,*, Alberto Bellin b

a Daniele Tonina, University of California Berkeley, and at USFS Rocky Mountain Research Station Boise, 322 East Front Street,

Suite 401, Boise ID 83702, United Statesb Alberto Bellin, Dipartimento di Ingegneria Civile ed Ambientale, Universita di Trento, via Mesiano 77, I-38050 Trento, Italy

Received 23 March 2007; received in revised form 17 July 2007; accepted 28 August 2007Available online 4 September 2007

Abstract

Pore-scale dispersion (PSD), aquifer heterogeneity, sampling volume, and source size influence solute concentrations of conservativetracers transported in heterogeneous porous formations. In this work, we developed a new set of analytical solutions for the concentra-tion ensemble mean, variance, and coefficient of variation (CV), which consider the effects of all these factors. We developed these modelsas generalizations of the first-order solutions in the log-conductivity variance of point concentration proposed by [Fiori A, Dagan G.Concentration fluctuations in aquifer transport: a rigorous first-order solution and applications. J Contam Hydrol 2000;45(1–2):139–163]. Our first-order solutions compare well with numerical simulations for small and moderate formation heterogeneity and from smallto large sampling and source volumes. However, their performance deteriorates for highly heterogeneous formations. Successively, weused our models to study the interplay among sampler size, source volume, and PSD. Our analysis shows a complex and important inter-action among these factors. Additionally, we show that the relative importance of these factors is also a function of plume age, of aquiferheterogeneity, and of the measurement location with respect to the mean plume center of gravity. We found that the concentrationmoments are chiefly controlled by the sampling volume with pore-scale dispersion playing a minor role at short times and for smallsource volumes. However, the effect of the source volume cannot be neglected when it is larger than the sampling volume. A differentbehavior occurs for long periods, which may be relevant for old contaminations, or for small injection volumes. In these cases, PSDcauses a significant dilution, which is reflected in the concentration statistics. Additionally, at the center of the mean plume, where highconcentrations are most likely to occur, we found that sampling volume and PSD are attenuating mechanisms for both concentrationensemble mean and coefficient of variation, except at very large source and sampler sizes, where the coefficient of variation increases withsampler size and PSD. Formation heterogeneity causes a faster reduction of the ensemble mean concentrations and a larger uncertaintyat the center of the mean plume. Therefore, our results highlight the importance of considering the combined effect of formation heter-ogeneity, exposure volume, PSD, source size, and measurement location in performing risk assessment.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Solute transport; Groundwater; Concentration moments; Stochastic methods

0309-1708/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.advwatres.2007.08.009

* Corresponding author. Tel.: +1 208 3734381; fax: +1 208 3734391.E-mail addresses: [email protected] (D. Tonina), Alberto.Bellin@

unitn.it (A. Bellin).

1. Introduction

A great effort has been spent in the last few decades forbetter understanding the spatial and temporal evolution ofsolute concentrations in heterogeneous porous formations.Important advances have been obtained in this field mostlyconsidering point concentrations, which are defined over a

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mailto:Alberto.Bellin@ unitn.it

340 D. Tonina, A. Bellin / Advances in Water Resources 31 (2008) 339–354

small but finite volume of size ‘. According to the contin-uum approach to flow in porous formations ‘ should besmaller than the typical scale of variability of the hydraulicconductivity (IY), but large enough to define macroscopicquantities such as porosity and hydraulic conductivity(see e.g., [3], ch. 1). Hence, when this property is satisfied,the continuum approach applies at scales larger than ‘and point values of physical quantities can be definedand treated with the mathematical machinery of contin-uum theory (see e.g., [2], ch. 1). However, a large disparityof sampling volumes, whose dimensions depend on the wayconcentrations are measured, is observed in applications.For instance, solute concentrations are usually measuredin observation wells, spanning several vertical integralscales of the log-conductivity, or in the effluent of extrac-tion wells, with a sampling volume that can be approxi-mated by their capture zone for continuous, or long-lasting, extractions. Therefore, point measurements arerare and the characteristic size of the sampling volume istypically much larger than ‘. In these cases, besides dilutionassociated with pore-scale dispersion (PSD), which mixessolute at scales smaller than ‘, measured solute concentra-tions are influenced by mixing of solute mass within thesampling volume.

An important case in which attenuation of solute con-centrations associated with the sampling volume shouldbe carefully evaluated is risk assessment, because samplingvolume influences solute concentrations dramatically at theexposure location. For instance, sampling volumes vary ifrisk analysis of exposure to a contaminant is assessed ina drinking water distribution system at the production wellor at the in-house taps. In the former case, the samplingvolume would be the volume of the well, whereas in the lat-ter case, a reasonable choice would be the aquifer volumewithin the well capture zone with travel time less or equalto the turnover time, tr = Vr/Q, of the storage facility,whose volume is Vr, and Q is the mean water dischargeextracted from the well. In both cases, the level of exposuredepends critically on the interplay between the spatial dis-tribution of the solute mass within the aquifer and the sizeof the sampling volume.

The impact of the sampling volume on the first twomoments of the solute concentration has been investigatedby Andricevic [1], who concluded that whereas both PSDand sampling volume attenuate solute concentrations, theirrelative importance depends on the elapsed time since injec-tion. In his derivation, he utilized closures developed in sol-ute transport in turbulent flows to quantify the correlationterms between concentration and velocity fields and a num-ber of additional assumptions in order to illustrate qualita-tively the impact of the sampling volume on theconcentration variance. In particular, he computed theconcentration variance by integrating, over the samplingvolume, the product of the variance of the point concentra-tion and the correlation function. The former has beenassumed constant over the sampling volume and the latterhas been assumed to decrease exponentially with the two-

point separation distance (see Eqs. (33), (34) and (35) of[1]). However, these solutions have been utilized by Andri-cevic to study the impact of the sampling volume qualita-tively on solute concentration, and they cannot be usedas a predictive tool in risk assessment because of the manyassumptions he adopted.

To overcome this difficulty and to further explore theimpact of PSD and sampling volume on the concentrationmoments, we developed new first-order solutions of thefirst two moments of volume-averaged solute concentra-tions. We started our development from the first-ordersolutions obtained by Fiori and Dagan [12] for the firstand second moments of point solute concentrations in het-erogeneous aquifers. In the remaining sections, first wegeneralize these solutions to include the effect of the sam-pling volume DV and then, after a thorough testing withaccurate numerical simulations, we use them to analyzethe effect of DV, source size, and PSD on the moments ofsolute concentrations.

2. First-order solutions for solute concentrations

Let us consider a tracer of constant concentration C0

released instantaneously within the volume V0, and assumethat the total mass M0 = n V0C0 is subdivided into Np non-interacting particles, each of them with the elementarymass Dm = n C0 V0/Np, where n indicates the formation’sporosity, which is considered constant through the aquifer(e.g., [7,14,16]). The point concentration C(x,t) expressesthe mass of solute per unit volume of fluid containedinto an elementary infinitesimal volume (DV! 0) centeredat x:

Cðx; tÞ ¼Z

V 0

daC0ðaÞd½x� Xtðt; aÞ� ð1Þ

where d is the Dirac delta function, and Xt(t;a) is the trajec-tory of the particle released at position x = a within V0:

x ¼ Xtðt; aÞ ¼ eXðt; aÞ þ Xb ð2Þ

Following Fiori and Dagan [12] in Eq. (2) the particle tra-jectory has been split into two components: the advectivedisplacement eX, which in principle can be obtained by solv-ing the following integro-differential equation deXðtÞ=dt ¼v½XtðtÞ�, where v is the local velocity, and the Browniancomponent Xb. The latter is a Wiener process with zeromean and variance matrix Xb,ii(t) = 2Dd,iit resulting froma constant diagonal dispersion tensor Dd,ii = aiijUj,i = 1, . . . ,N [1,12,20]. In the last two expressions, N is thespace dimensionality, aii is the pore-scale dispersivity, andU is the aquifer mean velocity, which for simplicity, butwithout lack of generality, is assumed aligned with the x1

direction. The Brownian motion is introduced to modelpore-scale dispersion, a process occurring at a scale smallerthan ‘, and from Eq. (2) can be expressed as the differencebetween the particle trajectory and the advective displace-ment (Xb ¼ Xt � eX).

D. Tonina, A. Bellin / Advances in Water Resources 31 (2008) 339–354 341

We define the resident concentration CDV for a volumeDV centered at x as the solute mass per unit of volume thatat time t is within DV:

CDV ðx; tÞ ¼1

DV

ZDV

dx

ZV 0

daC0ðaÞd½x� Xtðt; aÞ� ð3Þ

Accepting that a detailed mapping of the formationhydraulic properties cannot be obtained, we model thelog-conductivity Y = ln(K), where K is the local hydraulicconductivity, as a stationary random space function(RSF). Consequently to this choice, solute concentrationCDV is also modeled as a RSF with probability densityfunction that depends on the statistical properties of Y,through flow and transport equations.

The ensemble mean of CDV is obtained by taking the sta-tistical expectation of Eq. (3):

hCDV ðx; tÞi ¼1

DV

ZDV

dx

ZV 0

daC0ðaÞZ

deX Z dXb

� d½x� eXðt; aÞ � Xb�f ðeX;XbÞ ð4Þ

where f ðeX;XbÞ is the pdf of the particle trajectory. Becauseof the particle trajectory splitting into advective andBrownian components, f coincides with the multivariatejoint pdf of eX and Xb, which, according to the Bayes rule,is given by the product of the marginal pdf of eX and thepdf of Xb conditional to eX: f ðeX;XbÞ ¼ u0ðeXÞuCðXb j eXÞ.Fiori and Dagan [12] showed that at the first-order ofapproximation in r2

Y , advective and Brownian componentsof the total displacement are independent. This greatly sim-plifies the computation of f ðeX;XbÞ, which assumes the fol-lowing form:

f ðeX;XbÞ ¼YNi¼1

u0iðeX iÞ ! YN

i¼1

uiðX b;iÞ !

ð5Þ

where u0i and ui are the marginal pdfs of the ith componentof the advective and Brownian displacements, respectively.They are both normally distributed [8,12] with pdfs:

u0iðeX iÞ ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2pX iiðtÞp exp �ð

eX i � ai � Utdi1Þ2

2X iiðtÞ

" #ð6Þ

and

uiðX b;iÞ ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4pDd;iitp exp �

X 2b;i

4Dd;iit

" #ð7Þ

Furthermore, in Eq. (6) Xii i = 1, . . . ,N are the componentsof the displacement variance tensor X iiðtÞ ¼ heX 0iðt; aÞ�eX 0iðt; aÞi, with eX 0i ¼ eX i � heX ii representing the fluctuationof the advective component of the particle trajectoryaround its ensemble mean. Because of the first-orderapproximation, the components of the velocity field andof the corresponding advective displacement, which de-pends linearly on them, are also normally distributed.

Despite the great attention received in stochastic theo-ries the ensemble mean concentration, also in its volume-averaged form of Eq. (4), is a crude approximation of the

actual (unknown) solute concentration, as a number of nat-ural head gradient tracer tests have shown (e.g., [13]). Ameasure of the difference between actual and ensemblemean concentrations is provided by the coefficient ofvariation:

CV ðCDV Þ ¼rCDV

hCDV ið8Þ

which besides hCDVi requires the computation of the con-centration variance:

r2CDVðx; tÞ ¼ hC2

DV ðx; tÞi � hCDV ðx; tÞi2 ð9Þ

where the first term of the right hand side of Eq. (9) as-sumes the following form:

hC2DV ðx; tÞi ¼

1

DV 2

ZDV

dx0Z

DVdyhCDV ðx0; tÞCDV ðy; tÞi ð10Þ

After specializing CDV(x,t) of Eq. (3) to the locations x = x 0

and x = y within DV and substituting the resulting expres-sions into Eq. (10) we obtain:

hC2DV ðx; tÞi ¼

1

DV 2

ZDV

dx0Z

DVdy

ZV 0

da

ZV 0

db

ZdeX Z deY

� C0ðaÞC0ðbÞ/ðeX; eY; x0 � eX; y� eYÞ ð11Þ

In Eq. (11) / is the joint pdf of eXðt; aÞ, eYðt; bÞ, ðx0 � eXÞand ðy� eYÞ, where we used the condition x0 ¼ eX þ Xb

and y ¼ eY þ Yb for a particle originating in a and b,respectively, stemming from the property of the Dirac deltafunction in the Eq. (3). Owing to the independence ofadvective and Brownian displacements, / assumes the fol-lowing form:

/ðeX; eY; x� eX; y� eYÞ ¼ gðeX; eY; t; a; bÞuðx� eXÞ� uðy� eYÞ ð12Þ

where g is the joint pdf of the advective components of thetrajectory of two-particles simultaneously released at timet = 0 and positions x = a and x = b, respectively. We referto the work of Fiori and Dagan [12] for the discussion ofthe hypotheses underlying Eq. (12), while here we reporttheir conclusion that the first-order approximation of g ismultivariate normal:

gðeX; eY; t; a; bÞ ¼YNi¼1

giðeX i; eY i; t; ai; biÞ ð13Þ

where gi is the multivariate normal pdf of the ith compo-nents of eX and eY:

giðeX i; eY i; t; ai; biÞ

¼ 1

2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNiiðt; a� bÞ

p exp X iiðtÞðxi � ai � d1iUtÞ2hn

�2Ziiðt; a� bÞðxi � ai � d1iUtÞðyi � bi � d1iUtÞ

þX iiðtÞðyi � bi � d1iUtÞ2i� 2Niiðt; a� bÞ½ ��1

oð14Þ

Fig. 1. Sketch of the two-dimensional computational domain.


where Niiðt; a� bÞ ¼ X iiðtÞ2 � Ziiðt; a� bÞ2. Besides Xij,the two-particle trajectory covariances Zijðt; a� bÞ ¼heX0iðt; aÞeY 0jðt; bÞi are needed for computing / [12].

The relative importance of PSD with respect to advec-tion is measured by the Peclet number, which in naturalformations may assume different values in longitudinal,PeL = UIYh/Dd,11, and transverse horizontal, PeTh =UIYh/Dd,22 and vertical PeTv = UIYv/Dd,33, directions,where IYh and IYv are the horizontal and vertical log-con-ductivity integral scales, respectively. Because we addressthe problem in a two-dimensional domain we set PeT =PeTh, while PeTv is inconsequential. There are experimentalevidences suggesting that Pe = O(102) � O(103) [11] withthe smaller values observed in longitudinal direction. Inorder to explore the impact of PSD in the simulations wekept constant the transverse Peclet number to PeT = 1000and varied the longitudinal one such as to explore theimpact on concentration moments of different anisotropyratios RPe = PeT/PeL. PSD enters in Eq. (14) through Xii

and Zii, which are influenced by mixing between stream-lines. Streamline mass exchange in turn is controlled byPeT and is only slightly influenced by PeL (see [10,12] foran exhaustive treatment of this issue). Thus, the lattercan be neglected in the expression of Xii and Zii proposedby Fiori [10] and Fiori and Dagan [12], respectively.

The non-linear dependence of Zii on a and b preventsobtaining a closed form solution of Eq. (11). To overcomethis difficulty Fiori and Dagan [12] limited themselves toinjection volumes of characteristic size smaller than thelog-conductivity integral scale such that Zij(t;a � b) ffiZij(t; 0), allowing closed form solutions. Notwithstanding,we show in Section 4 that results obtained by using thissimplifying assumption are accurate well beyond the for-mal limit of small source volume.

The computations leading to the first-order solutions ofthe first two moments of the solute concentration CDV withsupport volume DV for a fully three-dimensional domainare summarized in the Appendix A.

3. Numerical simulations

In order to contain the computational effort withinacceptable limits we focus on a two-dimensional formationwith hydraulic log-conductivity modeled as a RSF withconstant mean hYi and variance r2

Y , and following an iso-tropic exponential covariance function:

CY ðrÞ ¼ r2Y exp½�r=IY �; ð15Þ

where r is the two-point separation distance and IY = IYh isthe log-conductivity integral scale.

We define the sampling volume DV as a rectangularsolid of square area DA, which represents the samplingarea, and height equal to the aquifer thickness, b. Similarly,the solute source is a volume V0 = A0b, where A0 is thesource area. Furthermore, as shown in Fig. 1, the samplingarea DA is square with side D, and the source area A0 is

rectangular with sides L1 and L2 in longitudinal and trans-verse directions, respectively.

To reproduce the variations of the hydraulic conductiv-ity accurately on the computational grid, we used a squaregrid with side fixed at 0.25IY, as suggested by Bellin et al.[5] and supported by Rubin et al. [17], who showed thatthe contribution of the wiped-out variability on solutespreading is negligible for block’s sizes smaller than0.25IY. Flow is solved by using the Galerkin’s finite elementmethod with triangular elements and linear shape func-tions. The grid’s block is divided into two triangles, whichshare the longest side and same hydraulic conductivity.Furthermore, the uniform grid’s spacing leads to elementsof same size, such that the maximum principle is respectedeverywhere within the domain, and velocity distribution isconsistent with Darcy’s law [15].

Transport following an instantaneous release of solutewith constant concentration C0 within the source volumeV0 is solved in a Lagrangian framework with the total sol-ute mass, M0 = nC0V0, partitioned into Np particles eachone of mass Dm = M0/Np. As customary, we assume thatparticles travel without deformation along the trajectoryof their centers of mass [18]:

X t;jðt; aÞ ¼ X t;jðt � Dt; aÞ þ V j Xtðt � Dt; aÞ½ �Dt

þ X b;jðtÞ ð16Þwhere X b;j ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2DtDd;jj

pej is the jth component of the

Brownian jump simulating diffusion by PSD; here Dd,jj isjth component of the diagonal dispersion tensor, and ej isa random variable normally distributed with zero meanand unitary variance. Furthermore, a is the location wherethe particle is released within the source volume V0, Vj(x) =Ud1j + uj(x) is the jth component of the local ‘‘Eulerian’’velocity field with constant mean U = (U, 0) and fluctuationu = (u1,u2), while d1j is the Kronecker’s delta.

Eq. (16) is applied recursively to all particles, and themean concentration of the solute within the sampling vol-ume DV = DAb centered at the position x is computed asfollows:

CDV ðx; tÞ ¼npðx; tÞDm

nDV¼ C0V 0

DVnpðx; tÞ

Np

ð17Þ


where np(x, t) is the number of particles that are within thehorizontal projection DA of the sampling volume DV attime t. According to this methodology, commonly knownas forward particle tracking (FPT), a particle is consideredeither inside, or outside of DV, depending on the positionof its center of mass. It can be shown that this procedureconverges to the exact solution of the transport equationwhen Np!1.

According to Eq. (17), particle tracking results in step-wise variations of the solute concentration with discreteincrement equal to DCDV,min = C0V0/(NpDV), which is alsothe minimum detectable concentration. Consequently, inorder to maintain constant DCDV,min, a reduction of theratio DV/V0 should be counterbalanced by an increase ofthe total number of particles (Np). Thus, an exceedinglylarge number of particles is needed in order to assesspoint sampling accurately when the source volume is largemaking FPT inefficient particularly for Monte Carlostudies.

In such cases, a valuable alternative is the backward par-ticle tracking (BPT) approach proposed by Vanderborght[19] and applied successfully in numerical simulations byCaroni and Fiorotto [6], who showed convergence of thismethod with the analytical solution of point concentrationsfor different degrees of pore-scale dispersion. The method-ology consists of two steps. A given number, nf, of soluteparticles are placed within the sampling volume DV cen-tered at x. Successively, each particle is moved backwardin time toward the source at each time step with two succes-sive displacements: a Brownian jump, modeled as:

X �j ðt � Dt; xÞ ¼ X t;jðt; xÞ � X b;j j ¼ 1; 2 ð18Þ

and an advective motion, based on the local velocityV = U + u at the position x = X* (t � Dt;x):

X t;jðt�Dt;xÞ ¼ X �j ðt�Dt; xÞ � V j½X�ðt�Dt; xÞ�Dt j¼ 1;2:

ð19Þ

If a particle changes grid element and hence velocity duringthe time interval Dt, its path is divided into as many sub-paths as the changes in velocity and the interval is updatedat each sub-step. Eqs. (18) and (19) are applied recursivelyto all particles. At the last step of the backward procedurethe position of all particles at t = 0 is identified and soluteconcentration is computed by the following equation:

Cðx; tÞ ¼ npðV 0; tÞnf

C0 ð20Þ

where np(V0, t) is the number of particles inside the sourcevolume at time t = 0, and C0 is the solute concentrationwithin the source. Eq. (20) considers the fact that onlythe particles ending their backward path within the sourcearea contribute to the solute concentration at time t andlocation x. In this scheme the minimum detectable concen-tration is given by: DCDV,min = C0/nf, corresponding to thecase when only one of the particles released within DV endsits journey backward in time within the source.

Notice that BPT and FPT are complementary method-ologies; for a given number of particles BPT is more accu-rate than FPT when DV� V0, while FPT is preferablewhen DV ’ V0, or larger. For this reason numerical resultspresented in this work are obtained either by using FPT orBPT depending on the ratio DV/V0.

4. Comparison of first-order and numerical solutions

In this section, we compare our first-order solutions ofhCDVi, r2

CDVand CV(CDV) for finite sampling volume with

numerical Monte Carlo simulations. We considered asquare source area with side L1 = L2 = L = 0.5IY andRPe = 10. Additionally, we adopted a minimum detectableconcentration of 6.2510�4C0 and 2000 Monte Carlo simu-lations (MC), which obtained convergence of the large timeCV(CDV).

Fig. 2a–c show hCDVi, r2CDV

and CV(CDV), respectively,at the center of mass of the mean plume in a weakly heter-ogeneous formation with r2

Y ¼ 0:2 for the following threesampling volumes: D = 0.2IY, 0.5IY and 1IY. In addition,the corresponding first-order solutions of Fiori and Dagan[12] for point concentrations are depicted by a dashed-dotline in each one of the three plots. Our first-order solutionsof both the ensemble mean and variance compare very wellwith the numerical simulations, as one can observe by com-paring lines and symbols in Fig. 2a and 2b, respectively.Furthermore, these figures show the impact of samplingvolume, which reduces the peak of both moments withrespect to point concentration solutions.

An important property of a plume is ergodicity, whichfacilitates data interpretation and modeling because in thissituation the ensemble mean concentration resembles theactual (observable) concentration. Strictly speaking ergo-dicity is reached when CV(CDV)! 0, a condition that,however, is hard to obtain, as clearly shown in Fig. 2c,which compares our first-order solution of CV(CDV) (Eq.(8)) with the Monte Carlo simulations. As expected, a lar-ger sampling volume results in a smaller CV(CDV), and ourfirst-order solution captures this feature accurately. Fig. 2calso shows that the sampling volume has a significantimpact on uncertainty affecting small plumes at short tointermediate times. At longer times since injection, theimpact of the sampling volume reduces and becomes negli-gible for t > 100IY/U when convergence to ergodicity iscontrolled by PSD. Later in the paper, we will explorethe reciprocal role of V0 and DV in obtaining an opera-tional ergodicity.

For DV = 0.2IY the ensemble mean concentrationobtained with our first-order model is in a good agreementwith numerical simulations and both show small differenceswith the first-order solution of Fiori and Dagan [12] forpoint concentrations. The three solutions show a slightlylarger difference when considering the concentration vari-ance of Fig. 2b, but reduces to a level comparable to thatof the ensemble mean when the coefficient of variation ofFig. 2c is considered. These results are in line with what

(a)

(b)

(c)

Fig. 2. First-order solutions of (a) hCDVi (Eq. (A.1)), (b) r2CDV

(Eq. (9)),and (c) CV(CDV) (Eq. (8)) at the center of mass of the mean plume (i.e., atx = (Ut,0)) are compared with numerical simulations (symbols) and thefirst-order solutions for point concentration of Fiori and Dagan [12](dashed-dot lines). Several sampling volumes are considered and in allcases, the source area is a square with side L = 0.5IY, r2

Y ¼ 0:2,PeT = 1000, and RPe = 10.


was previously observed by Bellin et al. [4], and suggestthat if we use CV as a measure of uncertainty then pointconcentrations can be operationally defined as the concen-

tration detected with a sampling device, whose size is smal-ler than 0.2IY.

Formation heterogeneity play an important role in sol-ute transport and in the performance of the first-order solu-tions. Consequently, we investigated its effect by repeatingthe previous simulations with increased r2

Y to 1. Numericaland first-order solutions are reported in Fig. 3, whichshows that our models perform well also for moderate het-erogeneity (r2

Y ¼ 1) and capture the sampling size mixingprocess, which is absent in Fiori and Dagan [12] solutions.Additionally, our results show a faster decline of theensemble mean concentration as a consequence of a stron-ger solute spreading due to the more complex spatial vari-ability of the velocity field than in the previous weaklyheterogeneous case. A closer inspection of Figs. 2b and3b reveals that a larger formation heterogeneity results inan earlier and higher peak of the concentration variance,followed by a faster decline (for instance compare r2

CDVval-

ues at time t = 10 tU/IY in Figs. 2b and 3b), due to aquicker decay of the ensemble mean concentrations.

However, the most important consequence of a larger r2Y

is that CV(CDV) increases dramatically and in turn also theuncertainty, as shown in Fig. 3c where the peak is roughlytwice than in Fig. 2c. It is also visible that the sampling vol-ume DV exerts a less intense attenuating effect on CV(CDV)than in the previous case.

So far we considered solute concentrations at the centerof mass of the mean plume. To test our solution in otherparts of the plume in Fig. 4 we analyze the evolution ofthe first two moments and the coefficient of variation ofsolute concentrations at the following two positions:x = (2.5IY,0) and x = (10IY,0), with the origin of the refer-ence system at the center of the source. These two positionsmay be assumed as representative of resident concentra-tions at short and intermediate distances from the source,respectively. In Fig. 4, our first-order solutions are com-pared with numerical simulations for weakly heterogenousformations, i.e. r2

Y ¼ 0:2 and for RPe = 1. Source dimen-sions are L1 = L2 = 1IY, and results are shown for bothD = 0.2IY and D = 1IY. At both distances from the source,our first-order solutions of hCDVi (Eq. (A.1)) and r2

CDV(Eq.

(9)) are in good agreement with numerical simulations withthe maximum difference observed for D = 1IY. The attenu-ating effect of the sampling volume on hCDVi is significantclose to the source, but reduces with the distance andbecomes small to negligible at intermediate distances. Sim-ilarly, this effect on r2

CDVreduces with distance, but less rap-

idly than for hCDVi. This different behavior of the first twomoments is reflected to CV(CDV) shown in Fig. 4c. Noticethat in Fig. 4b and a the reduction of r2

CDVis partially com-

pensated by the concomitant reduction of hCDVi, such thatonly a moderate reduction of CV(CD), and thus of theuncertainty, is associated with the significant enlargementof the sampling volume from D = 0.2IY to D = 1IY at bothdistances from the source (see Fig. 4c).

The attenuation effect of the sampling volume on hCDVireduces when larger values of r2

Y are used, as it is shown in

(a)

(b)

(c)


(Eq. (9)),and (c) CV(CDV), (Eq. (8)) at the center of mass of the mean plume (i.e., atx = (Ut,0)) are compared with numerical simulations (symbols) and thefirst-order solutions for point concentration of Fiori and Dagan [12](dashed-dot lines). Several sampling volumes are considered and in allcases, the source area is a square with side L = 0.5IY, r2

Y ¼ 1, PeT = 1000,and RPe = 10.

0 5 10 15

0 5 10 15

0 5 10 15

0.0

0.1

0.2

0.3

0.4

0.5

0.6 First-order, x = 2.5IY;Δ = 0.2IY First-order, x = 2.5IY;Δ = 1IY First-order, x = 10IY;Δ = 0.2 First-order, x = 10IY;Δ = 1 Numerical, x = 2.5IY;Δ = 0.2IY Numerical, x = 2.5IY;Δ = 1IY Numerical, x = 10IY;Δ = 0.2IY Numerical, x = 10IY;Δ = 1IY

<CΔV

>/C

0

tU/IY

0.00

0.05

0.10

0.15

0.20

σ CΔV

2 /C02

tU/IY

1

10

CV

[CΔV

]

tU/IY

(a)

(b)

(c)


(Eq. (9)),and (c) CV(CDV) (Eq. (8)) are compared with numerical simulations(symbols) at x = (2.5IY, 0) and x = (10IY,0) for several sampling volumes.In all cases, the source area is a square with side L = 1IY, r2

Y ¼ 0:2,PeT = 1000, and RPe = 1.


Fig. 5a, which is obtained with r2Y ¼ 1:2. As spreading

increases with heterogeneity, solute is dispersed over a lar-ger area faster resulting in two effects: a reduction of thesmoothing effect due to mixing within the sampling volumeand a lowering of the concentration peaks (c.f., Figs. 4a

and 5a). For instance, Fig. 5a (numerical results) showsthat the peak of hCDVi at x = (2.5IY,0) reduces by 6.7%when D is enlarged from 0.2IY to 1IY, while the sameenlargement produced a more pronounced reduction (i.e.,23.8%) for r2

Y ¼ 0:2 (see numerical results in Fig. 4). How-ever, a different behavior is observed for r2

CDV, with the

0 5 10 15 20 25 300.00

0.05

0.10

0.15

0.20 First-order, x = 2.5IY;Δ = 0.2IY First-order, x = 2.5IY;Δ = 1IYFirst-order, x = 10IY;Δ = 0.2First-order, x = 10IY; Δ = 1 Numerical, x = 2.5IY;Δ = 0.2IY Numerical, x = 2.5IY;Δ = 1IY Numerical, x = 10IY;Δ = 0.2IY Numerical, x = 10IY;Δ = 1IY

<CΔV

>/C

0

tU/IY

tU/IY

tU/IY

0 5 10 15 20 25 300.00

0.02

0.04

0.06

0.08

0.10

0.12

σCΔV

2 /C

02

0 5 10 15 20 25 30

1

10

CV

(CΔV

)

(a)

(b)

(c)


(Eq. (9)),and (c) CV(CDV) (Eq. (8)) are compared with numerical simulations(symbols) at x = (2.5IY,0) and x = (10IY,0) for several sampling volumes.In all cases, the source area is a square with side L = 1IY, r2

Y ¼ 1:2,PeT = 1000, and RPe = 1.


relative difference between the peaks for D = 0.2IY and 1IY

that increases from 72% to 76% when r2Y is changed from

0.2 to 1.2.On the other hand, for a given DV, the peaks of both

hCDVi and r2CDV

reduce progressively as r2Y increases, and

the aquifer becomes more heterogeneous. The drop of theconcentration variance with the distance from the sourceis much larger for r2

Y ¼ 1:2 than for r2Y ¼ 0:2, as emerging

from an accurate inspection of Figs. 4b and 5b. In fact, therelative reduction of r2

CDVbetween x = (2.5IY, 0) and

x = (10IY, 0) increases from 171% for r2Y ¼ 0:2 to 1000%

for r2Y ¼ 1:2. Solute spreading mechanisms, whose effi-

ciency increases with higher heterogeneity, could explainthis effect by anticipating the variance peak and conse-quently the variance decline as supported by comparingFigs. 2b and 3b. However, uncertainty depends on the coef-ficient of variation, which is analyzed subsequently.

The coefficient of variation shows a minimum for timescomparable with the mean travel time to the observationpoints. Comparison of Figs. 4c and 5c shows that the casewith a higher heterogeneity (r2

Y ¼ 1:2) has a higher mini-mum. Thus, a larger r2

Y leads to more uncertain predictionsof peak solute concentrations by their ensemble mean.Additionally, CV(CDV) increases for both shorter andlonger times, when the leading and trailing edges of theplume cross the observation point, as evidenced in Figs.4c and 5c (note the use of the logarithmic scale in the ordi-nate axis). The behavior of CV(CDV) along the trailing edgeof the plume, i.e. the tail of the resident concentrationcurve, differentiates the two cases. For long times sinceinjection, the coefficient of variation shows a more gentlerelative increase with time for r2

Y ¼ 1:2 than for r2Y ¼ 0:2.

As a consequence, the resident concentration curve showsa more extensive, yet less uncertain, tailing for r2

Y ¼ 1:2than for r2

Y ¼ 0:2.Furthermore, Fig. 5a and 5b shows that the match

between numerical and our first-order solutions of hCDViand r2

CDVdeteriorates as the aquifer becomes more hetero-

geneous (r2Y > 1). Numerical results for both hCDVi and

r2CDV

show smaller peaks and longer tailings than the corre-sponding first-order solutions, prolonging the persistenceof solute within the sampling volume. This effect is moreevident for observation points close to the source, wheretransport is not fully developed and the hypothesis thatthe particle’s trajectory is Gaussian does not hold [8].

5. Effects of PSD, sampling size, and source dimension

5.1. Point concentration

Fig. 6 shows the first two moments and the coefficient ofvariation of point concentrations versus the source volumeby increasing L from 0.1IY to 20IY with 0.1IY increments,and for the following two times: t = 10U/IY and t = 15U/IY. The source volume has a square horizontal projectionwith side L, centered at x = (10IY,0), and extends overthe entire aquifer’s thickness. Concentration measurementsare taken at the center of mass of the mean plume at timet = 10U/IY, whereas measurements at the latter time, whenthe center of the mean plume is ahead (downstream) of themonitoring location at x = (10IY, 0), represent sampling ina generic point within the mean plume. Solute mass was

Fig. 6. First-order and numerical solutions of ensemble mean ((a) and (b)), variance ((c) and (d)), and coefficient of variation ((e) and (f)) of solute pointconcentrations as a function of source size at x = (10IY,0) for t = 10IY/U (left column) and t = 15IY/U (right column), with r2

Y ¼ 0:2. Solutions obtainedin the absence of PSD (i.e., Pe!1) are compared with those obtained with PeT = 1000, and RPe = 10.


injected instantaneously and uniformly within the sourcevolume. The influence of PSD was analyzed by comparingthe concentration moments for Pe!1 with those forRPe = 10 in the same graph. In all cases considered inFig. 6, the first-order solutions of hCDVi (Eq. (A.1)) andr2

CDV(Eq. (9)), which as shown previously for DV! 0

resemble the solutions for point concentration obtained

by Fiori and Dagan [12], compare very well with the bench-marks provided by the numerical simulations.

What is immediately clear from Fig. 6a and b is thatPSD exerts a negligible influence on the ensemble meanconcentration at both times since injection as it was previ-ously reported (e.g., [8,10]). Nevertheless, it causes a signif-icant reduction of the concentration variance as shown in

0.10

0.20

0.30

0.400.50

0.60

0.70

0.80

0.90

1.0

2 4 6 8 10 12 14 16 18 20

1

2

3

4

5

Δ/I Y

Δ/I Y

L / IY

L / IY

0.10

0.20

0.30

0.40

0.50

0.60

0.700.80

0.90

2 4 6 8 10 12 14 16 18 20

1

2

3

4

5

(a)

(b)

Fig. 7. Contour plot of hCDVi as a function of source and samplingvolumes at x = (10IY, 0) for (a) t = 10 IY/U and (b) t = 15IY/U, withr2

Y ¼ 0:2. First-order solutions obtained with PeT = 1000, and RPe = 10(solid lines) are compared with those obtained in the absence of PSD(Pe!1, dashed lines).


Fig. 6c and d. The striking result shown in these two figuresis that the first-order solution is accurate beyond its formallimit of applicability for small source volume, i.e. L� IY,resulting from assuming Zii(t;a � b) ’ Zii(t; 0); i = 1,2 inEq. (18) of Fiori and Dagan [12].

As expected CV(C) decreases with source size reachingan operational (source size induced) ergodicity whenCV(C)! 0 for large L. However, this limit is not obtainedfor the same source size everywhere within the plume. Infact, CV(C) decreases to zero more rapidly at the centerof mass of the plume, where concentrations are expectedto be the highest, than in other positions, as shown inFig. 6e and f. Notice that the drop of the numerical solu-tion in Fig. 6e for L/IY > 10 is an artifact due to the impos-sibility of measuring concentrations below the detectablelimit associated with the elementary mass of the particle.We run a simulation with double number of particles andthe outcome shows that the numerical solution drops at ahigher L. Since we are not interested in analyzing situationswith CV smaller than the one where the drop occurs inFig. 6e we do not increase the number of particles, whichwould have caused a much higher computational effort.

Because in all the above comparisons with r2Y < 1 our

first-order solutions are in a good agreement with thebenchmark numerical simulations also with large sourcesizes, we investigate the interplay among source size, sam-pling volume, and PSD with our first-order solutions in theremaining of this paper without showing their numericalcounterparts.

5.2. Volume-averaged concentration

We analyzed the effect of both source and sampling vol-umes on the concentration moments at x = (10IY, 0) byincreasing D from 0.05IY to 5IY with 0.05IY incrementsand the source volume as described above. Source andsampling volumes have a square horizontal projection withside L and D, respectively, and both extend over the entireaquifer’s thickness. Like in the previous simulations, solutemass was injected instantaneously and uniformly withinthe source volume.

We present the influence of D and L on hCDVi atx = (10IY,0) in Fig. 7a and b, obtained for t = 10IY/Uand t = 150 IY/U, respectively. Additionally, in the samefigures we compare hCDVi solutions predicted with(RPe = 10) and without PSD (i.e., Pe!1). Source sizeis the major factor controlling hCDVi, as shown by thenearly vertical contour lines in Fig. 7a and b, while PSDhas a weak influence and sampling dimension plays a moreimportant, yet minor role. The latter result was expected,since the solutions for different D values coalesce att > 10IY/U in Figs. 2 and 3 and at distances of the orderof 10IY from the source in Figs. 4 and 5.

We can explain these results from a physical point ofview as follows. Dilution associated with PSD and mixingwithin the sampling volume both contribute to attenuatethe spatial variability of the solute concentration. How-

ever, since PSD acts at scales of the order of centimeters(the Darcy’s scale), its influence on hCDV i is marginal withrespect to mixing when D is larger than a few centimeters,i.e. a small fraction of IY, (D > ‘).

The concentration variance is expected to be much moresensitive to PSD, but the overwhelming effect of large mix-ing volumes may mask the PSD effect also for r2

CDV. Fig. 8a

shows our first-order solution of r2CDV

(Eq. (9)) with(RPe = 10) and without PSD (i.e., Pe!1) atx = (10IY, 0) and for t = 10IY/U. In addition, Fig. 8b repli-cates Fig. 8a but for t = 15IY/U. We observe that a changeof PSD to values typically encountered in applications (i.e.,Pe O(102) � O(103)) from the idealized case thatneglects PSD (i.e., Pe!1), causes the concentration var-iance to reduce noticeably at small DV and negligibly atlarge DV, as one would expect, because in the latter casemixing within the sampling volume exerts an overwhelmingsmoothing effect on the solute concentration with respectto PSD. However, source volume acts differently att = 10IY/U and t = 15IY/U, because higher variance valuesoccur at different locations in the D–L plane. This is the

0.0100.030 0.030

0.010

0.070

0.12

0.16

0.0100.010

0.030

0.070

0.080

2 4 6 8 10 12 14 16 18 20

2 4 6 8 10 12 14 16 18 20

1

2

3

4

5

0.040

0.080 0.0800.12

0.040

0.16

0.18

0.20

0.040

0.080 0.080

0.12

0.040

0.16

0.171

2

3

4

5

Δ/I Y

Δ/I Y

L/IY

L/IY

(a)

(b)

Fig. 8. Contour plot of r2CDV

as a function of source and sampling volumesat x = (10IY, 0) for (a) t = 10 IY/U and (b) t = 15IY/U, with r2

Y ¼ 0:2.First-order solutions obtained with PeT = 1000, and RPe = 10 (solid lines)are compared with those obtained in the absence of PSD (Pe!1, dashedlines).

2.0

1.0

0.80

0.60

0.40 0.20 0.10 0.010

1

2

3

4

5

10

5.0

3.0

2.0

1.0

0.400.20 0.10

5.0

3.0

2.0

1.0

0.400.20

0.10

2 10 12 14 16 18 20

1

2

3

4

5

2 4 6 8 10 12

2 10 12 14 16 18 202 4 6 8 10 12

1

2

3

4

5

L/IY

L/IY

Δ/I Y

Δ/I Y

(a)

(b)

Fig. 9. Contour plots of CV(CDV) as a function of source and samplingvolumes at x = (10IY,0) for (a) t = 10IY/U and (b) t = 15IY/U, withr2

Y ¼ 0:2. First-order solutions obtained with PeT = 1000, and RPe = 10(solid lines) are compared with those obtained in the absence of PSD(Pe!1, dashed lines).


consequence of the fact that the effect of PSD is more sig-nificant where the concentration gradients are large, andthis happens particularly in areas surrounding the positionswhere hCDVi ’ 0.5C0.

Furthermore, the most distinct feature of the concentra-tion variance contour plot is a ‘‘ridge’’ that follows the con-tour line hCDVi = 0.5C0 in the D–L plane of Fig. 7. Alongthis ridge, PSD plays a strong effect on the concentrationvariance, but it fades with sampling size. Because of thisridge, a given value of r2

CDVcan be obtained with several

combinations of D and L. Therefore, our results evidencethe interplay between source and sampling volume andstrengthen the importance of knowing both in order tocompute the concentration variance accurately.

Additionally, as shown in Fig. 9a and b, CV(CDV) iscontrolled by L and except for small source volumes (i.e.,L < 4IY), with a very limited effect of D. Uncertaintyreduces fast with sampling size at small source volumes,but at a rate that declines with increasing of source size.In addition, PSD exerts a noticeable effect on CV(CDV) atsmall sources, which, however, fades as the source volumegrows large. Additionally, we can observe sampler-induced

ergodicity at small sources and source-induced ergodicityin both plots.

5.3. Point versus volume-averaged concentrations

Because volume-averaged concentrations are the mostcommon in applications, it is important to know their devi-ation from point measurements, which in turn are the mostimportant for biological and ecological analysis. This isbecause microorganisms are in contact with local concen-trations, whose intensity may vary from volume-averagedmeasurements. A measure of this difference (between vol-ume-averaged and point concentrations) is provided bythe following two quantities: RC = hCDVi/hCi, andRCV = CV(CDV)/CV(C). Note that values of both ratioslarger than unity denote larger volume-averaged than pointquantities, whereas the inverse is true for values less than 1.

Fig. 10a depicts RC for t = 10U/IY at x = (10IY,0). Solidcontour lines are for finite Peclet values (RPe = 10) anddashed lines are obtained neglecting pore-scale dispersion(i.e., Pe!1). As expected, pore-scale dispersion exerts anegligible impact on RC, which is controlled by both L

and D. Notice that the smallest RC values are observed

1.0

0.99

0.97

0.95

0.90

0.80

0.70

0.60

0.50

0.40

0.30

2 4 6 8 10 12 14 16 18 20

1

2

3

4

5

Δ/I Y

L/IY

(a)

(b)

Fig. 10. Contour plots of RC at x = (10IY, 0) for (a) t = 10IY/U and (b)t = 15IY/U. First-order solutions obtained with PeT = 1000, and RPe = 10(solid lines) are compared with those obtained in the absence of PSD(Pe!1, dashed lines). In all cases r2

Y ¼ 0:2.

Zone 3 Zone 2 Zone 1 Zone 2 Zone 3

C

C

C

C

C small

small

small

large large∂∂x

Fig. 11. Sketch of the three zones characterizing the concentrationattenuation within the mean plume.


when a small plume is sampled with a large sampling vol-ume (the left upper corner of Fig. 10a), and that RC

increases much rapidly with L for small rather than largeD values. Furthermore, the ensemble mean of point andvolume-averaged concentrations show relative differencessmaller than 1% (i.e., RC P 0.99) for D < L/4, and smallerthan 20% for D < L. In other words, significant differencesbetween the ensemble mean of point and volume-averagedconcentrations are observed when D > L.

Fig. 10b shows RC for t = 15U/IY and all other quanti-ties unchanged. This figure shows that RC can assume val-ues either smaller or larger than 1 depending on thelocation of the measurement point with respect to the cen-ter of mass of the mean plume, and the interplay between Dand L. This behavior of RC measured at a generic pointwithin the mean plume can be attributed to the change inrate of spatial attenuation of hCi (Fig. 11). By consideringthe mean plume at a given time, one can observe that therate of attenuation of hCi first increases with distance fromthe center of mass, reaches the maximum value at interme-diate distances (zone 2 in Fig. 11), and then declines slowly

to zero as the distance from the center of mass grows large(zone 3 in Fig. 11). Based on these observations, we iden-tify three zones: zone 1, which denotes the volume sur-rounding the center of mass of the mean plume with theslowest attenuation rate and whose extension depends onL, zone 2, which contains the volume with the highest con-centration reduction, and zone 3, which comprises theremaining volume (Fig. 11). Therefore, RC may be eitherlarger or smaller than one depending on where the sam-pling volume falls in among these three zones. In particu-lar, RC > 1 when DV is sampling partially in zone 2 and3, while RC < 1 when DV is between zone 2 and 1 orentirely in zone 3. Furthermore, RC 1 when DV fallsentirely within zone 1, owing to the small variability ofthe solute concentration within this zone, or zone 2, dueto the constant decline of the solute concentration.

The streaking result shown in Fig. 10b is that hCDVi canbe larger than hCi when DV is large enough to collect sol-ute mass in zone 2 and 3. However, Fig. 10a shows that atthe center of mass of the mean plume hCDVi 6 hCi as onewould expect. Nevertheless, because this is the locationwith the highest probability of observing high concentra-tions, we conclude that the sampling volume introduces aconsiderable attenuation effect on the maximum concentra-tion with important consequence for risk analysis.

In Fig. 12, we investigated RC at the center of mass ofthe mean plume as a function of time for L = 10IY andtwo sampling volumes; one with D = 2IY and the otherwith D = 5IY corresponding to D/L = 0.2 and 0.5, respec-tively. Our results show that the influence of D varies withtime and is more pronounced for larger D. At early timeswhen the sampling volume is fully inside zone 1, RC = 1,but as time increases, macrodispersion acts reducing theextension of zone 1 such that RC reduces progressively.As larger portions of the sampling device fall in zone 2,RC reaches a minimum at intermediate travel times andthen it increases as effect of the homogenization of C overlarger volumes caused by solute spreading. At later times,RC reaches a relative maximum and then it decreases again.Note that the position of the minimum is independent onsampler size, which instead controls the absolute value.

100

101

102

103

104

105

0.95

0.96

0.97

0.98

0.99

1.00

x=(Ut,0); σY2 =0.2

L1=L

2=10 I

Y

RC

tU/IY

Δ/IY=2Δ/IY=5

Fig. 12. RC at the center of mass of the mean plume as a function of timeand for D = 2IY and 5IY. The source area is a square with side L = 10IY

and r2Y ¼ 0:2.


In the previous sections, we analyzed how uncertainty insolute concentration, quantified by CV(CDV), is influencedby D, PSD and source size. In Fig. 13 we present their com-bined effect on the ratio RCV = CV(CDV)/CV(C) betweenthe coefficients of variation of volume-averaged and pointconcentrations at x = (10IY,0) for both t = 10U/IY andt = 15U/IY. Fig. 13a and b are obtained with RPe = 10,

0.80

1.0 1.10.60

0.80

0.402.0

0.60

0.40

0.20

0.20 4.0

0.20

0.40

0.60

0.80

0.90

0.90

2 4 6 8 10 12 14 16 18 20

2 4 6 8 10 12 14 16 18 20

1

2

3

4

5

1

2

3

4

5

Δ/I Y

Δ/I Y

L/IY

(a) (

((c)

Fig. 13. Contour plots of RCV at x = (10IY,0) for (a) t = 10IY/U and Pe!1, ((d) t = 15IY/U, PeT = 1000, and RPe = 10. In all cases r2

Y ¼ 0:2.

while Fig. 13c and 13d are obtained in the absence ofPSD. Pair comparisons of Fig. 13a with c and b with d sug-gest that the influence of the sampling volume on RCV

depends on measurement location with respect to theexpected position of the center of mass besides on sourcesize and PSD.

Let us first focus on the case with pure advection(Pe!1). At the center of the mean plume (Fig. 13a),an increase of D causes RCV to decrease for small sources(let say L < 6IY), and to increase for large sources (let sayL > 6IY). The latter result suggests that uncertainty arehigher for volume-averaged than point concentrations atlarge source sizes. This has important consequences in riskanalysis, because point concentration uncertainty does notalways provide an upper limit. Therefore, it is important totake into account sampler and source sizes in assessing sol-ute exposure in applications. However, CV values are smallin these cases as shown in Fig. 9a.

On the other hand as shown in Fig. 13c, for a locationoutside the center of the mean plume (for t = 15IY/U, i.e.when the sampling volume is ahead of the expected posi-tion of the center of mass) an increase of D leads always(except for very large sampler and source sizes) to a reduc-tion of RCV, and this reduction attenuates with L.

In Fig. 13b and d we show the effect of PSD, which causesan attenuation of the differences between volume-averaged

1.1

1.0

2.0

0.80

0.60

0.40

4.0

1.0

0.20

1.0

0.90

0.80

0.600.40 1.1

2 4 6 8 10 12 14 16 18 20

2 4 6 8 10 12 14 16 18 20

1

2

3

4

5

L/IY

1

2

3

4

5

b)

d)

b) t = 10IY/U, PeT = 1000, and RPe = 10, (c) t = 15IY/U and Pe!1, and


and point concentration coefficients of variation. However,its effect on RCV is a function of sampler and source sizes asshown in Fig. 13b and d. PSD maintains RCV 1 at smallsampler sizes (e.g., D < 0.2IY), because of its smoothingmechanism that attenuates concentration spatial fluctua-tions. However, at larger D its effect may fade or reversedepending on source size. At small source sizes (e.g., L < 6IY), as the sampler size increases PSD effect fades disappear-ing at large sampling sizes, thereby leaving the sampler sizeas the major controlling factor. On the other hand, at largesource and large sampling volumes PSD causes an increaseddifference of RCV from unity and RCV grows larger than 1 asobserved previously. Therefore, the overall effect of PSD isof reducing the difference between CV(CDV) and CV(C),with the exception of large source and sampling volumes,for which PSD leads to larger uncertainty in volume aver-aged solute concentrations. (right-up corner of Fig. 13aand b and of c and d).

6. Conclusions

Ground water is an important resource and its carefulquality management depends on water concentration anal-ysis. However, concentration measurements are stronglydependent on the choice of the support volume (D) espe-cially as sampler size grows larger than the Darcy’s scale‘. Additionally, source size (L), pore-scale dispersion(PSD), and aquifer heterogeneity (r2

Y ) affect concentrationmoments.

Therefore, our first-order solutions of concentrationensemble mean, variance, and coefficient of variation areimportant tools in performing concentration analysis,because they account for all these factors. An importantoutcome of our analysis is that applicability of our modelsis not limited to small sources, despite their derivationunder the limiting condition of L! 0. Additionally, com-parison between numerical and first-order solutions showsthat our model performance is accurate up to moderateheterogeneity of the formation hydraulic properties (i.e.r2

Y ¼ 1) and then decreases with aquifer heterogeneity(r2

Y > 1:2).Our analysis reveals a complex interplay among sampler

size, source volume, and PSD, whose relative importance isalso a function of time since injection and aquifer heteroge-neity. Sampler volume smoothing effect on the concentra-tion first moment depends on the measurement locationwith respect to the mean plume center and the interplaybetween sampler and source volume. Whereas, mixingwithin the sampler’s volume always exerts an attenuatingeffect at the center of the mean plume, this is not alwaystrue in a generic point within the plume, where concentra-tions may be larger for volume-averaged than for pointmeasurements. However in all cases, sampling effect fadeswith source size and time since injection. This is acceleratedby an increase of formation heterogeneity, which attenu-ates the concentration first moment strongly, but increasesconcentration uncertainties at the center of the mean

plume. However, concentration uncertainties (CV) maydecrease with larger r2

Y at the plume fringe.Additionally, source size has a paramount influence on

the mixing effect of the sampler volume on concentrationuncertainty. At the center of the mean plume, the coeffi-cient of variation decreases rapidly with sampler volumeat small source size. However as source size increases, thiseffect fades and eventually reverses causing concentrationuncertainty to increase with sampling size. Notwithstand-ing, sampler volume causes a decrease of CV for other loca-tions around the center of the mean plume except for verylarge source and sampling volumes.

Furthermore, we suggest that sampler size mechanismmight be enhanced in a three-dimensional field as water iscaptured and mixed from different elevations. On the otherhand, its effect should decrease faster with distance or timesince injection, because as the plume spreads within theaquifer, it samples the formation heterogeneity more effi-ciently [9]. Another, factor, which we did not analyze in thiswork, but might affect the relative importance of PSD, sam-pler, and source volume is source shape, which has beenshown to influence concentration moments (e.g., [1]). How-ever, we hypothesize that source shape should not changethe outcome of our analysis, but it could probably antici-pate or postpone the onset of one factor’s influence.

Nevertheless, as time passes pore-scale dispersion, whichacts at a scale much smaller than the sampler volume mix-ing mechanism, starts influencing solute concentrationmoments. Whereas, PSD has a negligible effect on ensem-ble mean solute concentrations regardless of source andsampling dimensions, it attenuates concentration varianceand thus reduces uncertainty plaguing analyses based onthe evolution of ensemble mean concentrations. Addition-ally, PSD has another beneficial effect by reducing the dif-ference between point and volume-averaged CVs.However, its influence fades quickly with sampler size. Itis important to notice that PSD effect on the differencebetween point and volume-averaged concentration CVsreverses for very large source and sampler sizes causing vol-ume-averaged measurements to have higher uncertainty.

As last remarks, we summarize the following 4 findingsfor the zone around the plume center of mass, which com-prises the highest solute concentrations and thus bearimportant information in risk assessment:

(1) At short times since injection, and as long as D� L,the main factor controlling concentration distribu-tion is the sampling volume, with pore-scale disper-sion and source size both playing a minor role.

(2) Source size should be considered when it is of thesame order or smaller than the sampling size(L 6 D), and both sampling volume dimension andPSD play an equal important role.

(3) At long times since injection, which may be the casefor old contaminations, PSD introduces an apprecia-ble additional smoothing to the concentration, whilesampling volume plays a minor role.


(4) Increases of aquifer heterogeneity causes a reductionof the ensemble mean concentration, but an increasein uncertainty.

Acknowledgements

We would like to thank Daniel Fernandez-Garcia andfive anonymous reviewers for their insightful and carefulreviews. The second author acknowledges the support ofthe European Commission through the project AquaTerra(contract no. 505428).

Appendix A. Ensemble mean and variance of passive soluteconcentrations

We consider a two- or three-dimensional formation withhydraulic property variations described by a geostatisticalmodel of spatial variability epitomized through the mean,variance and covariance function of the hydraulic log-con-ductivity (or log-transmissivity in the case of a two-dimen-sional regional model). Pore-scale dispersion isparameterized through the longitudinal PeL = UIYh/Dd,11

and transverse horizontal PeTh = UIYh/Dd,22 and verticalPeTv = UIYv/Dd,33 Peclet numbers. For a two-dimensionalformation only the longitudinal and horizontal transversePeclet numbers are defined. Notice that our results arenot limited to these two models of spatial variability, andother geostatistical models can be utilized provided thatthe moments Xii and Zii are available.

By substituting Eq. (5) of the joint probability distribu-tion of eX and Xb into Eq. (4) we obtain, after integration,the following expression for hCDVi (Eq. (4)):

hCDV ðx; tÞiC0

¼YNi¼1

I iiðxi; tÞ ðA:1Þ

where the functions Iii, i = 1, . . . , N are given by

I iiðxi; tÞ ¼1

Li

1

4ð2xi � Di þ Li � 2di1tÞ½

�

� erfDi � Li � 2xi þ 2di1t

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2X t;iiðtÞ

p" #þ ðDi þ Li � 2xi þ 2di1tÞ

� erfDi þ Li � 2xi þ 2di1t


p" #� ð2xi þ Di � Li � 2di1tÞ

� erfDi � Li þ 2xi � 2di1t


p" #þ ð2xi þ Di þ Li � 2di1tÞ

� erfDi þ Li þ 2xi � 2di1t


p" ##þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiX t;iiðtÞ

2p

r

� � exp �ðDi � Li þ 2di1t � 2xiÞ8X t;iiðtÞ

� ��þ exp �ðDi þ Li þ 2di1t � 2xiÞ

8X t;iiðtÞ

� �

� exp �ðDi � Li � 2di1t þ 2xiÞ8X t;iiðtÞ

� �þ exp �ðDi þ Li � 2di1t þ 2xiÞ

8X t;iiðtÞ

� ��ðA:2Þ

In Eq. (A.2), Xt,ii(t) = Xii(t) + 2Dd,iit is the total dis-placement variance. Notice that as shown by Fiori [10]Xij is influenced by the transverse Peclet numbers (i.e. PeTh

and PeTv for the three-dimensional case and PeT = PeTh forthe two-dimensional case) and only slightly by PeL, whichis then neglected in the expression of Xii. The source andthe sampling volume have dimensions Li, and Di

i = 1,2,3, respectively, in the three-dimensional case. Inthe two-dimensional case the source is rectangular withsides Li, i = 1,2 and extending over the aquifer thicknessb. The sampling volume is also of thickness b with dimen-sions Di, i = 1,2 in the horizontal plane. As shown in Fig. 1the subscript ‘‘1’’ refers to quantities measured along themean flow direction. Furthermore, in Eq. (A.2) lengthsare dimensionless with respect to the log-conductivity inte-gral scale IYh, time t with respect to IYh/U, and particlesdisplacement variances Xt,ii with respect to I2

Yh. In theabsence of pore-scale dispersion Eq. (A.1) specialized tothe two-dimensional case resembles the first-order solutionobtained by Bellin et al. [4].

We consider now r2CDV

, which according to Eq. (9) canbe decomposed into two components. We also assume thatZii is independent from the releasing points of the particles.Under this simplifying assumption and after substitutingEq. (12) into Eq. (11), and integrating the resulting expres-sion, we obtain:

hC2DV ðx; tÞi ¼

C20QN

i¼1

Di

� �2

YNi¼1

Z xiþDi=2

xi�Di=2

dr

�Z Li=2

�Li=2

daiHðr; aiÞ ðA:3Þ

where the function H(r,ai) is given by

Hðr; aiÞ ¼1

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pX t;iiðtÞ

p exp �ðr � di1t � aiÞ2

2X t;iiðtÞ

" #� Giðci; xi þ Di=2Þ � Giðci; xi � Di=2Þ½�Giðai; xi þ Di=2Þ þ Giðai; xi � Di=2Þ� ðA:4Þ

with

Giðf; sÞ ¼e�ðf�sbiÞ2ffiffiffi

pp

bi

þ s� fbi

� �erfðsbi � fÞ ðA:5Þ

Finally, the functions ai, bi and ci are given by

ai ¼ biLi

2þ di1t þ r � di1t � aið Þ Ziiðt; 0Þ

X t;iiðtÞ

� �ðA:6Þ

bi ¼X t;iiðtÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2X t;iiðtÞðX t;iiðtÞ2 � Ziiðt; 0Þ2Þq ðA:7Þ

ci ¼ ai � Libi ðA:8Þ


Inspection of Eqs. (A.1) and (9) reveals that for vanish-ing DV, both hCDVi and r2

CDVconverge to the expressions

proposed by Fiori and Dagan [12].

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The influence of source size and sampling volume on the concentration pdf of conservative tracers released in heterogeneous formations